a mattress underwent the following changes in price before being removed from the store.\n|20%|markdown|\n|35…

a mattress underwent the following changes in price before being removed from the store.\n|20%|markdown|\n|35%|markup|\n|24%|markdown|\n|13%|markup|\n|31%|markdown|\nif the final price of the mattress was $227.83, what was the original price of the mattress, to the nearest dollar? round all dollar values to the nearest cent.\na. $356\nb. $312\nc. $251\nd. $146

a mattress underwent the following changes in price before being removed from the store.\n|20%|markdown|\n|35%|markup|\n|24%|markdown|\n|13%|markup|\n|31%|markdown|\nif the final price of the mattress was $227.83, what was the original price of the mattress, to the nearest dollar? round all dollar values to the nearest cent.\na. $356\nb. $312\nc. $251\nd. $146

Answer

Explanation:

Step1: Convert percentages to decimals

A markdown of $x%$ means multiplying by $(1 - \frac{x}{100})$ and a markup of $x%$ means multiplying by $(1+\frac{x}{100})$. $20%$ markdown: multiplier $m_1=1 - 0.20=0.80$ $35%$ markup: multiplier $m_2=1 + 0.35 = 1.35$ $24%$ markdown: multiplier $m_3=1 - 0.24=0.76$ $13%$ markup: multiplier $m_4=1 + 0.13 = 1.13$ $31%$ markdown: multiplier $m_5=1 - 0.31=0.69$

Step2: Let the original price be $P$.

The final - price $F$ is related to the original price by the equation $F = P\times m_1\times m_2\times m_3\times m_4\times m_5$. We know $F = 227.83$, and $m_1\times m_2\times m_3\times m_4\times m_5=0.80\times1.35\times0.76\times1.13\times0.69$. First, $0.80\times1.35 = 1.08$. Then, $1.08\times0.76=0.8208$. Next, $0.8208\times1.13 = 0.927504$. Finally, $0.927504\times0.69 = 0.640977$.

Step3: Solve for $P$.

Since $F = P\times0.640977$, then $P=\frac{F}{0.640977}$. Substitute $F = 227.83$ into the formula: $P=\frac{227.83}{0.640977}\approx355.43\approx356$.

Answer:

A. $356$